Pion interactions in the X(3872)
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(2) S. FLEMING, M. KUSUNOKI, T. MEHEN, AND U. VAN KOLCK PC. . The quantum numbers J 1 arise if the X3872 is a C , S-wave molecular bound state of D0 D 0 D 0 D0 . The possibility of a shallow molecular state is motivated by the proximity of the X3872 to the D0 D 0 threshold and naturally explains the large isospin violation observed in pion decays and the dominance of the D0 D 0 0 decay mode. The narrow width and the nonobservation of decays such as X ! c are highly unusual for a conventional charmonium state above the DD threshold. From the mass in Eq. (1) and the recent measurement of the D0 mass in Ref. [8] one infers a binding energy EX mD mD mX 0:6 0:6 MeV:. (4). This favors a bound-state interpretation of the X3872; however, because of the large uncertainty, the mass alone cannot rule out a resonance or ‘‘cusp’’ near the D0 D 0 threshold [21]. In this paper we will assume the X3872 is a molecular bound state, though our method can be extended to the case where the X3872 is a shallow resonance. For other interpretations, see Refs. [22 –37]. A recent review can be found in Ref. [38]. The interpretation as a DD molecule is particularly predictive because the small binding energy implies that the molecule has universal properties that are determined by the binding energy [39– 44]. The small binding energy can be further exploited through factorization formulae for production and decay rates of the X3872 [45,46]. Voloshin calculated the decays X ! D0 D 0 0 [39] and X ! D0 D 0 [40] using the universal wave function of the molecule. The main purpose of this paper is to consider the effect of 0 exchange on the properties of the X3872. Consider the one-pion-exchange contribution to D0 D 0 ! D0 D 0 scattering depicted in Fig. 1. This leads to an amplitude g2 ~ q~ ~ q~ ; 2f2 q~ 2 2. (5). where g is the D-meson axial (transition) coupling, f is the pion decay constant, ~ and ~ are the polarization vectors of the incoming and outgoing D mesons, respectively, and q~ is the momentum transfer. The scale appearing in the propagator denominator is given by 2 2 m2 , where is the D -D hyperfine splitting and m. FIG. 1. One-pion-exchange diagram for D0 D 0 ! D0 D 0 scattering. The single and double lines represent the spin-0 and spin-1 D mesons, respectively. The dashed line represents the 0 .. PHYSICAL REVIEW D 76, 034006 (2007). is the neutral pion mass. The hyperfine splitting, , appears in the pion propagator because the exchanged pion carries ~ Note that is energy q0 ’ as well as momentum q. anomalously small, 45 MeV, because of the nearness of 142 MeV and m 135 MeV. This suggests that pions generate anomalously long-range effects and should be included as explicit degrees of freedom in the description of the molecule, if the binding energy in Eq. (4) is not much smaller than its upper limit. The pion interactions in the D and D system were quantitatively analyzed using a one-pion-exchange potential model by Tornqvist [47], who actually predicted a DD bound state (deuson) with a mass close to the observed X3872. After the discovery of the X3872, Swanson [26] considered a potential model that includes both a one-pionexchange potential and a quark-exchange potential and found a weakly bound state in the S-wave JPC 1 channel. These authors worked in the isospin limit and used isospin-averaged pion masses and hyperfine splittings and obtained long-range Yukawa-like potentials [47]. Note that the effective mass term in the propagator in Eq. (5) has the opposite sign from what one typically obtains from meson exchange. This leads to a 0 -exchange potential in position space which is oscillatory rather than Yukawalike, as pointed out by Suzuki [2]. A central point of this paper is that the effect of 0 exchange can be dealt with using perturbation theory. Naive dimensional analysis of the relative size of twopion- and one-pion-exchange graphs yields the ratio 1 1 g2 MDD ; 20 10 4f2. (6). where MDD is the reduced mass of the D and D and we have set g 0:5–0:7 [48–50]. This is in contrast with twonucleon systems where a similar estimate yields [51,52] 1 g2A MN m ; 2 2 8f. (7). where gA 1:25 is the nucleon axial coupling and MN is the nucleon mass. A perturbative treatment of pions fails in the 3 S1 channel where iteration of the spin-tensor force yields large corrections at next-to-next-leading order (NNLO) [53,54]. This is in part due to the large expansion parameter in Eq. (7) and in part due to large numerical coefficients appearing in the NNLO calculation. The amplitude in Eq. (5) also gives rise to a spin-tensor force and one may worry that the perturbative treatment of pions will fail. However, even if large NNLO coefficients like those found in Refs. [53,54] appear in similar diagrams for the X3872, the expansion parameter in Eq. (6) is small enough that one can reasonably expect perturbation theory to work. In this paper, we derive an effective field theory of the D0 D 0 and D0 D 0 interacting with neutral pions near the D0 D 0 threshold. This theory is very similar in structure to. 034006-2.
(3) PION INTERACTIONS IN THE X3872. PHYSICAL REVIEW D 76, 034006 (2007). the Kaplan-Savage-Wise (KSW) theory of NN interactions in Refs. [51,52] where a leading-order (LO) contact interaction is summed to all orders in perturbation theory to produce a bound state at LO and pion exchange is treated perturbatively.1 A novel feature of the effective theory for the X3872 is that the hyperfine splitting of the D0 and D0 is only 7 MeV above the 0 mass and thus the pions are included as nonrelativistic particles. In this paper we focus on the decay X ! D0 D 0 0 . Our results are easily extended to X ! D0 D 0 . At LO our theory reproduces Voloshin’s calculations using effective-range theory [39,40]. We then compute the NLO corrections to the decay width. These include effective-range corrections as well as calculable nonanalytic corrections from 0 exchange. We find that nonanalytic calculable corrections from pion exchange are negligible and the NLO correction is dominated by contact interaction contributions. This paper is organized as follows. In Section II, we give the Lagrangian and discuss power counting in our theory. In Section III, we describe our calculation of the partial width X ! D0 D 0 0 . In Section IV, we summarize and conclude. Appendix A describes how our Lagrangian is derived by integrating out the scales m and from heavyhadron chiral perturbation theory (HHPT) [57–59]. Appendix B gives the results of evaluating the individual NLO diagrams for the decay amplitude and the wave function renormalization. While this work was being completed, a related preprint [60] appeared which analyzed the effects of light-meson exchange on a bound state of heavy mesons near a threemeson threshold. This work used a scalar-meson model and calculated the entire line shape of the resonance to second order in the heavy-light meson coupling. Our work is complementary to that of Ref. [60] in that we do not use a model but rather a Lagrangian that is directly relevant to the X3872 and we go to higher order in the heavy-light meson coupling, where renormalization requires the introduction of higher-derivative contact operators. On the other hand, we do not calculate the full line shape but work at the resonance peak where a Breit-Wigner is a suitable approximation. II. LAGRANGIAN AND POWER COUNTING The mass of the X3872 in Eq. (1) is extremely close to the D0 D 0 threshold. Assuming that the X3872 is a hadronic molecule whose constituents are a superposition of the D0 D 0 and D0 D 0 , the X3872 binding energy is given by Eq. (4). For reasons stated earlier, our calculations assume positive binding energy and a molecular interpretation of the X3872. The upper bound on the typical 1. A pionless effective theory of shallow nuclear bound states in which the leading nonderivative contact interaction is resummed to all orders was first proposed in Ref. [55]. For a similar theory of the X3872 see Ref. [56].. momentum of the D and D in the bound state is then
(4) 2MDD EX 1=2 48 MeV, where MDD is the reduced mass of the D0 and D 0 . For this binding momentum the typical velocity of the D and D is approximately vD ’ EX =2MDD 1=2 & 0:02, and both the D and D are clearly nonrelativistic. We will use nonrelativistic fields for the D and D . The pion degrees of freedom are also treated nonrelativistically. The maximum energy of the pion emitted in the decay X ! D0 D 0 0 is. E . m2X 4m2D m2 142 MeV; 2mX. (8). which is just 7 MeV above the 0 mass at 134.98 MeV. The maximum pion momentum is approximately 44 MeV, which is comparable to both the typical D-meson momentum, pD & 48 MeV, and the momentum scale appearing in the pion-exchange graph, ’ 45 MeV. Since the velocity of the pions is v p =m 0:34, a nonrelativistic treatment of the pion fields is valid. In this respect the treatment of pions differs from ordinary chiral perturbation theory or the NN theory of Refs. [51,52]. The effective Lagrangian includes the charm mesons, the anticharm mesons, and the pion fields. We denote the fields that annihilate the D0 , D 0 , D0 , D 0 , and 0 as D, D, D, D, and , respectively. To the order we are working we will not need diagrams with charged pions and charged D mesons so these are neglected in what follows. We construct an effective Lagrangian that is relevant for lowenergy S-wave DD scattering, where the initial and the final states are the C superposition of D0 D 0 and D0 D 0 : 1 jDD i
(5) p jD0 D 0 i jD0 D 0 i: 2. (9). An interpolating field with these quantum numbers will be used to calculate the properties of the X3872. We integrate out all momentum scales much larger than the momentum scale set by pD p . For D mesons this corresponds to kinetic energy & 1 MeV; for pions the kinetic energy is & 7 MeV. The hyperfine splitting and m should be treated as large compared to the typical energy scale in the theory. We start from the Lagrangian of HHPT [57–59], which describes the interactions of D and D mesons with Goldstone bosons, and integrate out the scales m and by rephasing fields to eliminate the large components of their energy. The method is similar to the rephasing used to remove the large mass from the energies of the fields in heavy-quark effective theory [61]. Details are given in Appendix A. The effective. 034006-3.
(6) S. FLEMING, M. KUSUNOKI, T. MEHEN, AND U. VAN KOLCK. PHYSICAL REVIEW D 76, 034006 (2007). Lagrangian is ~2 ~2 ~2 ~2 ~2 r r r r r D D y i@0 L Dy i@0 D Dy i@0 D D y i@0 D y i@0 2mD 2mD 2mD 2mD 2m g 1 C ~ D y D r ~ y H:c: 0 DD DD DD y DD p p DDy r 2 2f 2m C B1 1 $ 2 ~ H:c: . ; DD $ 2 D Dr y Dr y DD r H:c: p p DD DD D (10) 2 DD 16 2 2m where m ’ 7 MeV. Note$ that 2 ~ and ឈ r, 2 m2 2m . We use the notation r r ‘‘. ’’ in Eq. (10) denotes higher-order interactions. The pion decay constant is f 132 MeV with our choice of normalization. Notice that, since we are only interested in a C superposition of the D0 D 0 and D0 D 0 defined in Eq. (9), contact interactions are written p in terms of the DD= combination of fields DD 2. Because is a nonrelativistic field, annihilates and y creates 0 quanta, so that the Lagrangian in Eq. (10) allows D0 ! D0 0 and D0 0 ! D0 transitions and forbids D0 0 ! D0 and D0 ! D0 0 . Therefore in this effective field theory the only channels that appear are In amplitudes with external D 0 D0 D 0 D0 and D0 D 0 p0 . pions, we must multiply by 2m because of the normalization of the nonrelativistic pion fields. In the X3872 ! 0 0 decay diagrams, this will cancel the factors of D0p D 1= 2m in the axial coupling and in the term proportional to B1 in Eq. (10). Other channels can of course couple to the X3872. The three-body channels D D 0 and D D 0 are above the X3872 by only 2:8 0:6 MeV and 3:0 0:6 MeV, respectively. These channels can only appear as virtual intermediate states in X3872 decay and self-energy graphs that contain at least two-pion exchanges. These graphs are NNLO and therefore do not appear at the order we are working.2 The D D threshold lies 8.7 MeV above the X3872, and we may integrate out these states because they lie outside the range of the effective theory. If kept in the theory, this intermediate state would also only appear at NNLO. One may worry about other nearby thresholds, especially J= and J= ! which are only 1:4 1:1 MeV and 8:2 1:0 MeV, respectively, above the X3872. The J= channel has a much smaller energy gap than the others. However, one should take into account that the magnitude of the complex energy gap includes the width of the , =2 73 MeV, so the J= channel can be safely integrated out [46]. A higher-precision analysis of the X3872 may need to include these thresholds explicitly, especially if one wishes to describe the decays 2. This assumes that the interpolating field for the X3872 is DD, i.e. is constructed from neutral D-meson fields / DD only. Since physical results should not depend on the choice of interpolating field, we are free to make this choice.. X ! J= and X ! J= 0 . We leave this to future work. Matching onto HHPT yields the D0 , D0 , and 0 kinetic terms as well as the axial D0 -D0 -0 coupling. The coupling constant, g, is determined from data on the decays of D mesons. The CLEO measurements of the D width yield g 0:59 0:07 at tree level [48,49]. A NLO analysis of D decays in Ref. [62] yields g 0:270:06 0:03 . A more recent analysis [50] obtains g 0:61 at tree level and g 0:660:53 at NLO, where the number outside parentheses refers to the result when virtual lowlying even-parity charmed mesons are included in the loop calculations and the number in parentheses refers to the result obtained when these states are integrated out. The uncertainty in the NLO extraction of g is estimated to be 20%. We will use g 0:6 0:1 in this paper. The remaining terms in Eq. (10) with coefficients C0 , C2 , and B1 are contact interactions that are not obtained from matching HHPT but must also be included. They incorporate effects that come from shorter distance scales than the scale coming from 0 exchange. We have only included operators needed to the order we are working. C0 and C2 mediate D0 D 0 D 0 D0 scattering in the C , S-wave channel and have zero and two derivatives, respectively. B1 mediates a transition between D0 D 0 D 0 D0 in the C , S-wave channel to a state with a D0 , D 0 , and 0 . In our power counting, pD pD p Q, and we calculate amplitudes in an expansion in powers of Q. Since the D0 , D0 , and 0 are all nonrelativistic, ED ED E Q2 , so the propagators of all particles are order Q2 . Loop integrations are order Q5 . The D0 -D0 -0 axial coupling is order Q. In the exchange diagram of Fig. 1, one can drop the energy dependence p in the pion propagator. The factors of 2m from the vertices cancel the factors of 1=2m in the momentumdependent term in the pion propagator and combine with to give 2m 2 , reproducing the expression in Eq. (5). The pion-exchange amplitude is order Q0 as is easily seen from Eq. (5). Only counting powers of momentum, the Feynman rules for the terms in the Lagrangian with coefficients C0 , C2 , and B1 are naively of order Q0 , Q2 , and Q1 , respectively. However, with this power counting the theory is perturbative and cannot produce a bound state. Instead we will treat. 034006-4.
(7) PION INTERACTIONS IN THE X3872. PHYSICAL REVIEW D 76, 034006 (2007). C0 nonperturbatively, along the lines of Refs. [51,55], and sum diagrams with C0 to all orders. At LO, using the power divergence subtraction (PDS) scheme one then finds [51] C0 . 2 1 ; MDD PDS. (11). where PDS is the dimensional-regularization parameter.3 Taking PDS of order Q we find C0 is order Q1 which justifies its resummation. In PDS, the coefficient C2 is order Q2 as is B1 , as we shall see below. No other short-distance operators are needed for our NLO calculation of X ! D0 D 0 0 . Feynman diagrams with C2 and B1 first contribute to X ! D0 D 0 0 at NLO. In addition to expanding in Q, we will make one more approximation in the NLO calculation of X ! D0 D 0 0 . In many cases it greatly simplifies calculations to expand in m =mD 0:07. This is an approximation we will perform when evaluating loop diagrams. It is not systematized in our power counting scheme. As emphasized earlier, the perturbative character of pion exchange depends on the smallness of the parameter appearing in Eq. (6). Our effective theory can be used even if dimensionless parameters conspire to render pion exchange nonperturbative, but in this case one-pion exchange would have to be resummed as done in the NN system [63].. ellipsis represents terms that are less important in the resonance region, E EX . We can define a function E, where iEx represents the C0 -irreducible graphs contributing to GE. Our definition of E is similar to the function defined in Appendix A of Ref. [64]. In terms of E, GE is GE . Since the real part of the denominator must vanish at E EX , we have 1 C0 ReEX 0, and expanding about E EX we obtain for GE i1=C0 E EX Re0 EX ImEX C0 E EX Re0 EX iC0 ImEX i 2 C0 E EX Re0 EX iC20 ImEX i (14) ; C0. GE . where 0 d=dE. From Eq. (14), we immediately see that ZE . III. DECAY RATE FOR X3872 ! D0 D 0 0 Here we describe our method for calculating the width of the X3872 resonance. We consider the following twopoint function of interpolating fields Xi D0 D 0i p 0 0i D D = 2 for the X3872 with the spin index i: GE ij . Z. d4 xeiEt h0jTXi xXj 0j0i. i ij. ZEX ...; E EX i=2. iE i ReE ImE. : 1 C0 E 1 C0 ReE iC0 ImE (13). (12). where EX is the binding energy of the X3872 and the. 1 ; C20 Re0 EX . . 2 ImEX : Re0 EX . (15). The function 2 Im corresponds to the square of the decay diagrams. The LO decay and wave function renormalization diagrams are shown in Figs. 2 and 3, respectively. The NLO decay diagrams are shown in Figs. 4 and 5, and diagrams for the NLO wave function renormalization are shown in Fig. 6. It is interesting to compare the result of evaluating the loop diagrams and taking the imaginary part with direct evaluation of the decay diagrams. Consider, for example, the evaluation of the two-loop diagram in Fig. 6(a) in Appendix B. The result of evaluating the graph is. g2 1 PDS 82D Z dD q Z dD l 1 1 D D 2 2m 2 2 2 2 q0 EX =2 q =2mD i q0 EX =2 q2 =2mD i 2f q li q lj 1 1 : (16) l0 EX =2 l2 =2mD i l0 EX =2 l2 =2mD i q0 l0 q l2 =2m i. Fig: 6a i. We perform the energy integrals by contour integration, taking the poles of the D-meson propagators. This yields 3. The dimensional-regularization parameter is usually denoted but we use a different symbol here to avoid confusion with the scale appearing in pion exchange.. 034006-5.
(8) S. FLEMING, M. KUSUNOKI, T. MEHEN, AND U. VAN KOLCK. PHYSICAL REVIEW D 76, 034006 (2007). g2 PDS 82D Z dD1 q Z dD1 l 1 1 Fig : 6a i 2 2 2D1 2D1 EX q2 =2MDD i EX l2 =2MDD i 2f q li q lj 2m EX l2 =2mD q l2 2 i 82D Z D1 Z D1 g2 d q d l 1 1 i 2 2MDD 2 PDS D1 D1 2 2 2 2 2 2 q i l 2 i 2f . . q2 =2mD . q li q lj : m 2 =MDD q2 =mD l2 =mD q l2 2 i. (17). The first two propagators that come from the D mesons clearly scale as Q2 . The last two terms in the pion propagator denominator, q l2 2 , scale as Q2 while the other terms scale as m =mD Q2 . Since m =mD 0:07 is comparable to our expansion parameter in Eq. (6), these terms can be systematically dropped. The neglected terms come from the pion kinetic energy, and in dropping them we are treating the pions in the potential approximation [65]. The final answer is then 82D Z D1 Z D1 q li q lj g2 d q d l 1 1 Fig : 6a i 2 2MDD 2 PDS D1 D1 2 2 2 2 2 2 2 q i l i q l2 2 i 2f g2 ij MDD 2 1 1 PDS log i 2 ; (18) PDS 2 2 4^ 2 2 2 i 2f 3 ^ 1=4 ln E =2. We have used where 1=4 three-dimensional rotational invariance to replace q li q lj with q l2 ij =3 and use the PDS scheme to evaluate the remaining scalar integrals. There is one instance when dropping m =mD corrections is not appropriate. To see this consider evaluating the imaginary part of Fig. 6(a) by evaluating the cut diagram. The cut runs through the D-meson and pion propagators. In the cut diagrams, these propagators are replaced with functions. For the D-meson propagators integrating over the functions is equivalent to taking the pole using contour integration. So the cut diagram is obtained from Eq. (17) simply by replacing the pion propagator with the corresponding function. Doing this and making the substitutions q ! pD and l ! pD so that q l pD pD p , one obtains Z d3 pD d3 p g2 1 1 D 2 2M j~ p~ j2 2 DD 2 2 2 5 pD pD 2 2f 2 m 2 m 2 m 2 2 2 p p p : (19) MDD mD D mD D. This clearly reproduces the interference term in Voloshin’s effective-range calculation of X ! D0 D 0 0 [39]. The function in Eq. (19) imposes the constraint on the phase space due to energy conservation. Dropping m =mD -suppressed terms in the function corresponds to neglecting the final-state D-meson’s kinetic energy and would leave the integrals over their momentum unconstrained. Clearly this is not a good approximation. Physically, it is also clear that the on-shell propagating pion in the final state cannot be treated in the potential approximation. Therefore, in evaluating ImEX we will calculate the decay amplitudes for the diagrams and integrate over the physical three-body phase. In diagrams with virtual pions, we drop the kinetic energy so the pions are potential.4 In the virtual diagrams this approximation is valid up to Om =mD corrections. Since our expansion parameter is expected to be 0.05– 0.1, making this approximation in the virtual NLO graphs induces an error of the same size as the NNLO correction. The LO decay diagram is shown in Fig. 2. The D propagator scales as 1=Q2 and the axial coupling scales as Q so the LO diagram is order Q1 . We show only one diagram, but there are two channels related by C-conjugation that are implied. It is straightforward to evaluate these diagrams and obtain g MDD p~ ~ pD ! pD : i (20) f p2D 2 X The LO contribution to the wave function diagram is shown in Fig. 3. The graph is OQ and therefore 4. FIG. 2. LO diagram for decay rate.. For further discussion on the role of recoil corrections, see Ref. [66].. 034006-6.
(9) PION INTERACTIONS IN THE X3872. PHYSICAL REVIEW D 76, 034006 (2007). FIG. 3. LO diagram for calculating wave function renormalization.. Re0 EX is OQ1 . The result of evaluating this graph and taking the derivative is M2 (21) Re 0LO DD : 2 The LO decay diagram in Fig. 2 is OQ1 so the leading contribution to ImEX from Fig. 2 is OQ2 . Dividing by the LO wave function renormalization which is OQ1 one sees that the leading contribution to the decay rate is OQ1 . The result reproduces Voloshin’s calculation of X ! D0 D 0 0 [39]: 2 dLO g2 1 1 2 ~ ~. 2 p. : X p2D 2 p2D 2 dp2D dp2D 323 f2 (22). (a). (b). FIG. 4. NLO diagrams for the decay rate involving pion exchange.. The NLO corrections to the decay rate are suppressed by one power of Q. They come from graphs, shown in Figs. 4 and 5, with one additional pion exchange or one insertion of C2 and B1 . These coefficients scale as Q2 . NLO contributions to the wave function renormalization are down by one power of Q as well. These contributions are given by the two-loop self-energy diagrams involving pion exchange or an insertion of C2 shown in Fig. 6. The results for individual diagrams are given in Appendix B. The final expression for the NLO differential rate is. dNLO dLO g2 MDD 42 2 MDD PDS 2 2 2 1 C2 PDS 6f2 dp2D dp2D dpD dpD 42 2 g gMDD 1 1 2 ~ ~ C B p. 2 PDS 1 PDS PDS X f 163 f p2D 2 p2D 2 g4 MDD Reh1 pD 1 1 2 Reh1 pD ~ ~ p. X 643 f4 p2D 2 p2D 2 p2D 2 p2D 2 g4 MDD Reh2 pD 1 1 ~ ~ ~ ~ ~ ~ p p p. . . p p ! p : X D X D D D 643 f4 p2D 2 p2D 2 p2D 2 The functions h1 p and h2 p are given in Appendix B. The first line in Eq. (23) is a multiplicative correction to the LO decay rate. Note that in the absence of pions. C2 PDS . 2 r0 1 ; MDD 2 PDS 2. C2 (a). (b). where r0 is the effective range. The term proportional to C2 in the first line of Eq. (23) reproduces the expected correction from the effective-range theory, in which the leading correction involving r0 comes from the modification of the normalization of the wave function:. (24) ER r. B1. FIG. 5. NLO diagrams for the decay rate which involve contact interaction.. (23). s r e. : 41 r0 r. (25). The second line in Eq. (23) is the interference between a short-distance local coupling of the X to the D0 D 0 0 state and the LO amplitude. Note that the coefficient of this term scales as 1=PDS and disappears if one takes PDS ! 1, confirming the short-distance nature of the contribution. The final terms are nonanalytic corrections due to pion exchange. These contributions turn out to give a very small ( 1%) contribution to the decay rate, so the. 034006-7.
(10) S. FLEMING, M. KUSUNOKI, T. MEHEN, AND U. VAN KOLCK. (a). (b) FIG. 6.. (c). contributions. Measurements of the X mass and partial decay width into D0 D 0 0 can naturally be explained within a molecular picture if the corresponding point in Fig. 7 falls within, or—due to higher orders—close to, this band. Values far outside the band can be accommodated only if short-range parameters or higher-order effects are anomalously large. In either case the appeal of our framework would be strongly diminished. IV. SUMMARY. (26) where is a dimensionless parameter we expect to be of order unity. Figure 7 shows the partial width X ! D0 D 0 0 as a function of the binding energy. The central solid line is the LO result. We use the central value for the tree-level extraction of the D-meson axial coupling, g 0:6. The band in Fig. 7 shows the NLO rate with the parameters r0 and varied between 1 ; 100 MeV. 1 1:. (27). As stated earlier the nonanalytic calculable corrections from pion exchange in Eq. (23) give negligible corrections. The band is dominated entirely by the contact interaction 60. Γ [keV]. 50 40 30 20 10 0.2. 0.4. 0.6. C2. NLO diagrams for calculating wave function renormalization.. NLO correction is entirely dominated by the contact interaction contributions. We will parametrize C2 according to Eq. (24), where r0 is to be interpreted as the short-distance contribution to the effective range. Since we have integrated out the scales m and , it is reasonable to take r0 100 MeV1 . We will parametrize gMDD C2 PDS B1 PDS PDS ; f 100 MeV3. 0 r0 . PHYSICAL REVIEW D 76, 034006 (2007). EX [MeV]. 0.8. 1. 1.2. FIG. 7. Decay rate for X ! D0 D 0 0 as a function of EX . We use g 0:6. The central solid line corresponds to the LO prediction. The band is the result of the NLO calculation when the parameters r0 and are varied in the ranges 0 r0 100 MeV1 and 1 1.. In this paper we have developed an effective field theory of nonrelativistic pions and D mesons that can be used to describe the properties of the X3872, assuming it is a weakly bound state of D0 D 0 and D0 D 0 with anomalously small binding energy. Because of an accidental cancellation between the D-meson hyperfine splitting and the mass of the 0 , pion exchange is characterized by a smaller scale than is typically the case in nuclear physics. This relatively small scale and the small axial coupling in the D-meson system (compared to the nucleon’s axial coupling) combine to make the corrections from 0 -meson exchange amenable to perturbation theory. This justifies the application of a theory similar to that proposed by Kaplan, Savage, and Wise for low-energy NN interactions [51,52], in which a leading-order contact interaction is resummed to all orders to produce the bound state, and pion exchange and higher-derivative contact interactions are treated within perturbation theory. This theory reproduces at leading order the calculation of X ! D0 D 0 0 by Voloshin [39] which exploits the universal behavior of the DD wave function in the limit of small binding energy. Effective-range corrections as well as other corrections from short-distance scales are encoded in higher-dimension contact operators in the theory. These corrections turn out to completely dominate nonanalytic calculable corrections from 0 exchange. Varying these coefficients within ranges determined by naturalness allows us to estimate the size of corrections to the leadingorder calculations of Voloshin. While it is somewhat disappointing that the nonanalytic calculable corrections from 0 exchange are so small that an experimental test of this aspect of the theory seems unlikely in the foreseeable future, the smallness of these corrections confirms one of the main points of this work, namely, that pion exchange can be dealt with using perturbation theory.. 034006-8.
(11) PION INTERACTIONS IN THE X3872. PHYSICAL REVIEW D 76, 034006 (2007). A naive estimate of the size of the NNLO corrections based on the expansion parameter in Eq. (6) is 1% or smaller. It is important to remember that in conventional nuclear physics, large corrections come from graphs with two or more pion exchanges in the 3 S1 channel, which first arise at NNLO. The two-pion-exchange graphs at NNLO come with large coefficients, 5, which ruin the perturbative expansion of KSW for two-nucleon systems [54]. In our case similar size coefficients in two-pion-exchange graphs should not ruin perturbation theory since even with a large coefficient 5, they would only be expected to be 5% or smaller. It would be interesting to perform the NNLO calculation to check this. A NNLO correction of 5% would dominate the nonanalytic NLO contribution but would be smaller than the uncertainty in the contact interaction contribution, indicating convergence of the expansion. In the unlikely case that pion exchange is nonperturbative, it can be resummed as done in nuclear physics [63]. It is straightforward to extend the analysis of this paper to other X3872 decay and production processes, such as X ! D0 D 0 or X ! J= ! J= . Coupling to J= and J= ! channels can be incorporated by including these degrees of freedom explicitly in the theory and coupling them to D0 D 0 D0 D 0 via contact interactions. It would be interesting to calculate 0 exchange to other decays of the X3872 to see if these corrections lead to any interesting observable effects. It would also be interesting to use data or theoretical calculations to fix some of the counterterms appearing in the theory so as render calculations in this paper more predictive. ACKNOWLEDGMENTS We would like to thank E. Braaten and C. Hanhart for useful discussions. This work was supported in part by the U.S. Department of Energy under Grant Nos.DE-FG0206ER41449 (S. F. and M. K.), DE-FG02-05ER41368 (T. M.), DE-FG02-05ER41376 (T. M.), DE-AC0584ER40150 (T. M.), and DE-FG02-04ER41338 (U. v. K.), by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (U. v. K.), and by Brazil’s FAPESP under a Visiting Professor grant (U. v. K.). We would also like to thank the hospitality of the University of Arizona (T. M.), the Center for Theoretical Physics at MIT (T. M.), the Kernfysisch Versneller Instituut at Rijksuniversiteit Groningen (U. v. K.), the Instituto de Fı´sica Teo´rica of the Universidade Estadual Paulista (U. v. K.), and the Instituto de Fı´sica of the Universidade de Sa˜o Paulo (U. v. K.), where part of this work was completed. APPENDIX A: DERIVING THE EFFECTIVE LAGRANGIAN FROM HHPT Heavy-hadron chiral perturbation theory (HHPT) [57– 59] can be used to derive the low-energy effective. Lagrangian for D mesons, D mesons, and pions relevant to the X3872. We begin with the two-component HHPT Lagrangian introduced in Ref. [67], L TrH y iD0 H g TrH y H A TrH y H; (A1) 4 where are the Pauli matrices, mH mH , H D. 3 ~ D, and A r=f O . Here D is a heavy vector field, D is a heavy pseudoscalar field, and is the pion field, ! p1 0 2 : (A2) p12 0 . Evaluating the traces in Eq. (A1) we obtain 3 D L 2Dy iD0 D 2Dy iD0 4 4 2gDy AD Dy D A 2igDy D A: (A3) Since we wish to describe a bound state of two heavy mesons the power counting of HHPT in powers of 1=mH is inappropriate. Instead we need to power count in the relative velocity v 1 of the heavy mesons. The kinetic energy which is subleading in 1=mH is leading in v, and as a consequence we must include the kinetic term in our Lagrangian, ~2 r L 2Dy iD0 D 2mD 4 ~2 3 r D 2Dy iD0 4 2mD 2gDy AD Dy D A 2igDy D A: (A4) We now rescale the heavy-meson fields 1 fD; Dg ! p ei3t=4 fD; Dg; 2. (A5). which gives ~2 ~2 r r y y L D iD0 D D iD0 D 2mD 2mD gDy AD Dy D A igDy D A: (A6) Since we are only interested in those terms involving D0 , D0 , and 0 we keep only these fields in the Lagrangian: ~2 ~2 r r y y L D iD0 D D iD0 D 2mD 2mD g ~ 0 D Dy D r ~ 0 p Dy r 2f g ~ 0: i p Dy D r (A7) 2f. 034006-9.
(12) S. FLEMING, M. KUSUNOKI, T. MEHEN, AND U. VAN KOLCK. Next the kinetic term for the pion is derived from the chiral Lagrangian f2 f2 Tr@ @ y B0 TrM y 8 4 1 1 @ 0 @ 0 m2 0 2 self-interactions 2 2 1 0 @2 m2 0 . ; (A8) 2 where exp2i=f , M diagmu ; md is the quarkmass matrix, and B0 is a constant. The pion self-interaction terms are not needed at the order we are working so they are dropped, and we add the pion kinetic term to Eq. (A7) to obtain our Lagrangian. However, this Lagrangian still includes the large scales m and , which must be integrated out of the theory. Since we are interested in a nonrelativistic theory of pions we are justified in splitting the pion fields into creation and annihilation operators 0 ^ ^ y . In addition we rescale the meson fields to make the large scales explicit: L . 1 ^ p eim t ; 2m. 1 ^ p eim t y ; 2m y. im t. D!e. (A9). D:. PHYSICAL REVIEW D 76, 034006 (2007). The terms above that include a large phase factor can be integrated out. In addition to modifying the pion propagator the field redefinition in Eq. (A9) modifies the kinetic term for D, ~2 ~2 r r y D i@0 D ! D i@0 D; 2mD 2mD (A11) y. where m ’ 7 MeV. Note that after the field rescaling the last term in Eq. (A7) contains a phase factor eim t , and can be dropped. Finally, we obtain the kinetic terms and axial coupling appearing in the Lagrangian of Eq. (10) by another rephasing of the D and fields, D ! ei t D;. ! ei t :. (A12). This just shifts the residual mass from the D kinetic term to the kinetic term. This last step is not essential, however it is convenient. The remaining terms in Eq. (10) are shortdistance interactions allowed by power counting and the symmetries of the theory.. The pion kinetic term can then be expanded: 1 ~ 2 m2 0 L 0 @20 r 2 1 ~ 2 2im @0. fy 2im @0 @20 r 4m ~ 2 y e2im t 2im @0 @2 r ~ 2 @2 r 0. APPENDIX B: NLO DIAGRAMS The NLO decay diagrams involving pion exchange are shown in Fig. 4. The result for Fig. 4(a) is. 0. i. ~ 2 y g e2im t y 2im @0 @20 r ~2 r y i@0 2m higher-order relativistic corrections:. h1 p . Z1. (A10). 2 PDS h1 pD p~ ~ X 3 h2 pD p~ D ~ X p~ D p~ pD ! pD ;. 2 g3 MDD 8f3 p2D 2 . (B1). where the functions h1 p and h2 p are given by. q dx p2 x2 p2 2 2 x 2 i. 0. p2 2 2 p2 2 2 2 4p2 2 1 2p tan. 2 3 2 p 2 2 4p 8p i h2 p . Z1 0. . p2 2 2 2 4p2 2 2 p2 p2 2 2 i ln 2 2 4p 16p3 p2. (B2). x2 dx p p2 x2 p2 2 2 x 2 i. 5p2 32 32 3p2 2 2 2 4p2 2 1 2p 3 tan i 4 p2 2 2 4 p2 2 2 4p 4p 8p5. i. 3p2 2 2 2 4p2 2 2 p2 ln 2 ; p2 16p5. 034006-10. (B3).
(13) PION INTERACTIONS IN THE X3872. PHYSICAL REVIEW D 76, 034006 (2007). The NLO diagrams with the counterterms C2 and B1 are shown in Fig. 5. The result for Fig. 5(a) is. and the result for Fig. 4(b) is . 2 g3 MDD 3 p~ ~ X pD ! pD : 12f3 p2D 2 2. (B4) iC2 PDS . Note that Fig. 4(b) includes, as a subgraph, the one-loop D self-energy contribution. In the PDS scheme the selfenergy graph has a linear divergence which gives an additive renormalization to the residual mass term of the D . At tree level, we performed a field redefinition which moved the residual mass term from the kinetic term of the D to the kinetic term of the pion through a field redefinition. In order that loop corrections do not reintroduce a residual mass for the D , we introduce a counterterm ct Dy D which is defined to cancel the residual mass term at D y D, each order in perturbation theory. At one-loop order, ct g2 2 PDS =24f2 . This linearly divergent contribution to the self-energy also appears in Fig. 6(b) and is canceled by an insertion of the residual mass counterterm in a DD bubble (not shown in the figure).. [1] M. B. Voloshin, in Proceedings of 4th Flavor Physics and CP Violation Conference (FPCP 2006), Vancouver, British Columbia, Canada, 2006 (unpublished). [2] M. Suzuki, Phys. Rev. D 72, 114013 (2005). [3] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, 262001 (2003). [4] D. Acosta et al. (CDF II Collaboration), Phys. Rev. Lett. 93, 072001 (2004). [5] V. M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 93, 162002 (2004). [6] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 71, 071103 (2005). [7] S. L. Olsen (Belle Collaboration), Int. J. Mod. Phys. A 20, 240 (2005). [8] C. Cawlfield et al. (CLEO Collaboration), Phys. Rev. Lett. 98 092002 (2007). [9] K. Abe et al. (Belle Collaboration), arXiv:hep-ex/ 0505037. [10] G. Gokhroo et al. (Belle Collaboration), Phys. Rev. Lett. 97, 162002 (2006). [11] G. B. Mohanty (BABAR Collaboration), arXiv:hep-ex/ 0509039. [12] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 96, 052002 (2006). [13] K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 93, 051803 (2004). [14] K. Abe et al. (Belle Collaboration), arXiv:hep-ex/ 0408116. [15] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 93, 041801 (2004). [16] C. Z. Yuan, X. H. Mo, and P. Wang, Phys. Lett. B 579, 74 (2004). [17] S. Dobbs et al. (CLEO Collaboration), Phys. Rev. Lett. 94, 032004 (2005).. 2 2 2 gMDD PDS pD p~ ~ X 4f p2D 2. pD ! pD ;. (B5). and the result for Fig. 5(b) is iB1 PDS . MDD PDS p~ ~ X : 2. (B6). Finally, we show the NLO wave function renormalization diagrams in Fig. 6. The NLO contribution to Re0 EX from the graphs in Fig. 6 is 3 g2 MDD PDS 22 122 f2 42 2 C2 PDS . 3 MDD PDS 2 PDS : 22. (B7). [18] A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 96, 102002 (2006). [19] K. Abe et al. (Belle Collaboration), arXiv:hep-ex/ 0505038. [20] A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 98 132002 (2007). [21] D. V. Bugg, Phys. Lett. B 598, 8 (2004). [22] T. Barnes and S. Godfrey, Phys. Rev. D 69, 054008 (2004). [23] E. J. Eichten, K. Lane, and C. Quigg, Phys. Rev. D 69, 094019 (2004). [24] N. A. Tornqvist, Phys. Lett. B 590, 209 (2004). [25] C. Y. Wong, Phys. Rev. C 69, 055202 (2004). [26] E. S. Swanson, Phys. Lett. B 588, 189 (2004). [27] F. E. Close and S. Godfrey, Phys. Lett. B 574, 210 (2003). [28] B. A. Li, Phys. Lett. B 605, 306 (2005). [29] K. K. Seth, Phys. Lett. B 612, 1 (2005). [30] L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Phys. Rev. D 71, 014028 (2005). [31] F. S. Navarra and M. Nielsen, Phys. Lett. B 639, 272 (2006). [32] M. Ishida, S. Ishida, and T. Maeda, arXiv:hep-ph/0509212. [33] J. Vijande, F. Fernandez, and A. Valcarce, Int. J. Mod. Phys. A 20, 702 (2005). [34] G. Bauer, Int. J. Mod. Phys. A 21, 959 (2006). [35] J. D. Kim, S. K. Lee, and J. B. Choi, arXiv:nucl-th/ 0601048. [36] E. S. Swanson, Phys. Rep. 429, 243 (2006). [37] P. Colangelo, F. De Fazio, and S. Nicotri, arXiv:hep-ph/ 0701052. [38] S. L. Zhu, arXiv:hep-ph/0703225. [39] M. B. Voloshin, Phys. Lett. B 579, 316 (2004). [40] M. B. Voloshin, Int. J. Mod. Phys. A 21, 1239 (2006). [41] E. Braaten and M. Kusunoki, Phys. Rev. D 69, 074005 (2004).. 034006-11.
(14) S. FLEMING, M. KUSUNOKI, T. MEHEN, AND U. VAN KOLCK [42] E. Braaten and M. Kusunoki, Phys. Rev. D 69, 114012 (2004). [43] E. Braaten, M. Kusunoki, and S. Nussinov, Phys. Rev. Lett. 93, 162001 (2004). [44] E. Braaten and M. Kusunoki, Phys. Rev. D 71, 074005 (2005). [45] E. Braaten and M. Kusunoki, Phys. Rev. D 72, 014012 (2005). [46] E. Braaten and M. Kusunoki, Phys. Rev. D 72, 054022 (2005). [47] N. A. Tornqvist, Z. Phys. C 61, 525 (1994). [48] S. Ahmed et al. (CLEO Collaboration), Phys. Rev. Lett. 87, 251801 (2001). [49] A. Anastassov et al. (CLEO Collaboration), Phys. Rev. D 65, 032003 (2002). [50] S. Fajfer and J. Kamenik, Phys. Rev. D 74, 074023 (2006). [51] D. B. Kaplan, M. J. Savage, and M. B. Wise, Phys. Lett. B 424, 390 (1998). [52] D. B. Kaplan, M. J. Savage, and M. B. Wise, Nucl. Phys. B534, 329 (1998). [53] S. Fleming, T. Mehen, and I. W. Stewart, Phys. Rev. C 61, 044005 (2000). [54] S. Fleming, T. Mehen, and I. W. Stewart, Nucl. Phys. A677, 313 (2000).. PHYSICAL REVIEW D 76, 034006 (2007). [55] U. van Kolck, Nucl. Phys. A645, 273 (1999). [56] M. T. AlFiky, F. Gabbiani, and A. A. Petrov, Phys. Lett. B 640, 238 (2006). [57] M. B. Wise, Phys. Rev. D 45, R2188 (1992). [58] G. Burdman and J. F. Donoghue, Phys. Lett. B 280, 287 (1992). [59] T. M. Yan, H. Y. Cheng, C. Y. Cheung, G. L. Lin, Y. C. Lin, and H. L. Yu, Phys. Rev. D 46, 1148 (1992); 55, 5851(E) (1997). [60] E. Braaten, M. Lu, and J. Lee, arXiv:hep-ph/0702128. [61] A. V. Manohar and M. B. Wise, Heavy Quark Physics, Cambridge Monographs on Particle Physics, Nuclear Physics, and Cosmology Vol. 10 (Cambridge University Press, New York, 2000), p. 58. [62] I. W. Stewart, Nucl. Phys. B529, 62 (1998). [63] A. Nogga, R. G. E. Timmermans, and U. van Kolck, Phys. Rev. C 72, 054006 (2005). [64] D. B. Kaplan, M. J. Savage, and M. B. Wise, Phys. Rev. C 59, 617 (1999). [65] T. Mehen and I. W. Stewart, Nucl. Phys. A665, 164 (2000). [66] V. Baru, C. Hanhart, A. E. Kudryavtsev, and U. G. Meissner, Phys. Lett. B 589, 118 (2004). [67] J. Hu and T. Mehen, Phys. Rev. D 73, 054003 (2006).. 034006-12.
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